Search Results (7)

A novel approach to the quantum version of $\kappa$-entropy that incorporates it into the conceptual, mathematical and operational framework of quantum computation is put forward. Various alternative expressions stemming from its definition emphasizing computational and algorithmic aspects are worked out: First, for the case of canonical Gibbs states, it is shown that $\kappa$-entropy is cast in the form of an expectation value for an observable that is determined. Also, an operational method named “the two-temperatures protocol” is introduced that provides a way to obtain the $\kappa$-entropy in terms of the partition functions of two auxiliary Gibbs states with temperatures $\kappa$-shifted above, the hot-system, and $\kappa$-shifted below, the cold-system, with respect to the original system temperature. That protocol provides physical procedures for evaluating entropy for any $\kappa$. Second, two novel additional ways of expressing the $\kappa$-entropy are further introduced. One determined by a non-negativity definite quantum channel, with Kraus-like operator sum representation and its extension to a unitary dilation via a qubit ancilla. Another given as a simulation of the $\kappa$-entropy via the quantum circuit of a generalized version of the Hadamard test. Third, a simple inter-relation of the von Neumann entropy and the quantum $\kappa$-entropy is worked out and a bound of their difference is evaluated and interpreted. Also the effect on the $\kappa$-entropy of quantum noise, implemented as a random unitary quantum channel acting in the system’s density matrix, is addressed and a bound on the entropy, depending on the spectral properties of the noisy channel and the system’s density matrix, is evaluated. The results obtained amount to a quantum computational tool-box for the $\kappa$-entropy that enhances its applicability in practical problems. Full article

Quantum Computing in the Noisy Intermediate-Scale Quantum (NISQ) era has shown promising applications in machine learning, optimization, and cryptography. Despite these progresses, challenges persist due to system noise, errors, and decoherence. These system noises complicate the simulation of quantum systems. The depolarization channel is a standard tool for simulating a quantum system’s noise. However, modeling such noise for practical applications is computationally expensive when we have limited hardware resources, as is the case in the NISQ era. This work proposes a modified representation for a single-qubit depolarization channel. Our modified channel uses two Kraus operators based only on X and Z Pauli matrices. Our approach reduces the computational complexity from six to four matrix multiplications per channel execution. Experiments on a Quantum Machine Learning (QML) model on the Iris dataset across various circuit depths and depolarization rates validate that our approach maintains the model’s accuracy while improving efficiency. This simplified noise model enables more scalable simulations of quantum circuits under depolarization, advancing capabilities in the NISQ era. Full article

We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups. Full article

We generalize an approach to studying the quantum state tomography (QST) of open systems in terms of the dynamical map in Kraus representation within the framework of dynamic generation of informationally complete positive operator-valued measures. As applications, we use the generalized approach to theoretically study the QST of qubit systems in the presence of nonequilibrium environments which exhibit nonstationary and non-Markovian random telegraph noise statistical properties. We derive the time-dependent measurement operators for the quantum state reconstruction of the single qubit and two-qubit systems in terms of the polarization operator basis. It is shown that the behavior of the time-dependent measurement operators is closely associated with the dynamical map of the qubit systems. Full article

We theoretically study the non-Markovian disentanglement dynamics of a two-qubit system coupled to nonequilibrium environments with nonstationary and non-Markovian random telegraph noise statistical properties. The reduced density matrix of the two-qubit system can be expressed as the Kraus representation in terms of the tensor products of the single qubit Kraus operators. We derive the relation between the entanglement and nonlocality of the two-qubit system which are both closely associated with the decoherence function. We identify the threshold values of the decoherence function to ensure the existences of the concurrence and nonlocal quantum correlations for an arbitrary evolution time when the two-qubit system is initially prepared in the composite Bell states and the Werner states, respectively. It is shown that the environmental nonequilibrium feature can suppress the disentanglement dynamics and reduce the entanglement revivals in non-Markovian dynamics regime. In addition, the environmental nonequilibrium feature can enhance the nonlocality of the two-qubit system. Moreover, the entanglement sudden death and rebirth phenomena and the transition between quantum and classical nonlocalities closely depend on the parameters of the initial states and the environmental parameters in nonequilibrium environments. Full article

We show that an interaction between a harmonic oscillator and a two-level test mass (TLTM) mediated by a local operations and classical communication (LOCC) channel produces a signature that in (D. Carney et al., PRX Quantum 2, 030330 (2021)) is claimed to be exclusively reserved for channels that can transmit quantum information. We provide an explicit example based on a measurement-and-feedback channel, explain where the proof of Carney et al. fails, discuss to what degree setups of this type can test the nature of the gravitational interaction and remark on some fundamental implications that an LOCC model of gravity may have in black hole physics. Full article

We consider three types of entities for quantum measurements. In order of generality, these types are observables, instruments and measurement models. If $\alpha$ and $\beta$ are entities, we define what it means for $\alpha$ to be a part of $\beta$. This relationship is essentially equivalent to $\alpha$ being a function of $\beta$ and in this case $\beta$ can be employed to measure $\alpha$. We then use the concept to define the coexistence of entities and study its properties. A crucial role is played by a map $\widehat{\alpha }$ which takes an entity of a certain type to one of a lower type. For example, if $\mathcal{I}$ is an instrument, then $\widehat{\mathcal{I}}$ is the unique observable measured by $\mathcal{I}$. Composite systems are discussed next. These are constructed by taking the tensor product of the Hilbert spaces of the systems being combined. Composites of the three types of measurements and their parts are studied. Reductions in types to their local components are discussed. We also consider sequential products of measurements. Specific examples of Lüders, Kraus and trivial instruments are used to illustrate various concepts. We only consider finite-dimensional systems in this article. Finally, we mention the role of symmetry representations for groups using quantum channels. Full article